          What is it: A Chi-square test is any statistical hypothesis test in which the test statistic has a chi-square distribution when the null hypothesis is true, or any in which the probability distribution of the test statistic (assuming the null hypothesis is true) can be made to approximate a chi-square distribution as closely as desired by making the sample size large enough. The Chi-square Test is a statistical test which consists of three different types of analysis 1) Goodness of fit, 2) Test for Homogeneity, 3) Test of Independence.

• The Test for Goodness of fit determines if the sample under analysis was drawn from a population that follows some specified distribution.
• The Test for Homogeneity answers the proposition that several populations are homogeneous with respect to some characteristic.
• The Test for independence (one of the most frequent uses of Chi-square is for testing the null hypothesis that two criteria of classification, when applied to a population of subjects are independent. If they are not independent then there is an association between them.

Some examples of Chi-squared tests:

• Pearson's chi-square test, also known as the Chi-square goodness-of-fit test or Chi-square test for independence. When mentioned without any modifiers or without other precluding context, this test is usually understood.
• Yates' Chi-square test, also known as Yates' correction for continuity.
• Mantel-Haenszel chi-square test.
• Linear-by-linear association Chi-square test.
• The portmanteau test in time-series analysis, testing for the presence of autocorrelation
• Likelihood-ratio tests in general statistical modelling, for testing whether there is evidence of the need to move from a simple model to a more complicated one (where the simple model is nested within the complicated one).

Why use it: Chi Square is the most popular discrete data hypothesis testing method. The other primary use of the chi-square test is to examine whether two variables are independent or not. What does it mean to be independent, in this sense? It means that the two factors are not related.

Where to use it: The Chi-square test is perfect for count data such as Pass / Fail or Accept / Reject.  If you have processes or inspectors that you suspect are performing differently a Chi-square test is the perfect way to confirm the answer.

When to use it: Generally speaking, the Chi-square test is a statistical test used to examine differences with categorical variables. The chi-square test is used in two similar but distinct circumstances:

1. for estimating how closely an observed distribution matches an expected distribution - we'll refer to this as the goodness-of-fit test
2. for estimating whether two random variables are independent.

How to use it: The key idea of the Chi-square test is a comparison of observed and expected values. How many of something were expected and how many were observed in some process.

Calculating a goodness-of-fit test with Chi-square:

1. Establish hypotheses
2. Calculate expected values for each cell of the table.
3. Calculate Chi-square statistic. Doing so requires knowing:
1. The number of observations
2. Observed values
4. Assess significance level. Doing so requires knowing the number of degrees of freedom
5. Finally, decide whether to accept or reject the null hypothesis.

Important Notes: It is important to keep in mind that the Chi-square test only tests whether two variables are independent. It cannot address questions of which is greater or less.

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